Multistate Treatments of the Electronic Coupling in DonorâBridge

Article pubs.acs.org/JPCA

Multistate Treatments of the Electronic Coupling in Donor−Bridge− Acceptor Systems: Insights and Caveats from a Simple Model Robert J. Cave* Department of Chemistry, Harvey Mudd College, Claremont, California 91711, United States

Marshall D. Newton* Department of Chemistry, Brookhaven National Laboratory, Upton, New York 11973, United States S Supporting Information *

ABSTRACT: We use a simple one-dimensional delta function electronic structure model (df m) to investigate the results of a pair of multistate diabatization techniques (i.e., based on n states, with n ≥ 2) for linear DBA and DBBA (donor−bridge−acceptor) electron-transfer systems. In particular, we focus on the physical meaning of the couplings obtained from multistate methods and their relationship to two-state (n = 2) coupling elements. On the basis of the simple dfm approach, which allows exact as well as finite basis set treatment and has no many-electron effects, we conclude that for orthogonal diabatic states, it is difficult to assign clear physical significance to multistate matrix elements for coupling beyond nearest-neighbor contacts. The implications of these results for more complex multistate many-electron treatments are discussed. It is emphasized that physically meaningful coupling elements must involve states that are orthogonal, either explicitly or implicitly.

1. INTRODUCTION Born−Oppenheimer (BO) electronic states form the basis for much of our understanding of chemical processes.1 These BO (or adiabatic) states diagonalize the electronic Hamiltonian at a fixed nuclear geometry, and the spectrum of these states yields fixed-nuclei energies for the ground and excited electronic states of a system. In the event that the adiabatic potential energy surfaces are well-separated from each other, the reactivity of a system often can be understood based on motion on a single surface (i.e., an adiabatic process). Even when the adiabatic states are energetically proximate, it is still possible to use them as a basis, augmented with nuclear states associated with each potential (i.e., vibronic states (electronic vibrational state products)), to describe chemical reactivity in the so-called “nonadiabatic” regime.2,3 Despite the simplicity of the adiabatic description of the ground and excited electronic states of molecules, many applications are more naturally suited to use of so-called “diabatic” basis states.3 Diabatic states do not usually diagonalize the fixed-nuclei (BO) electronic Hamiltonian at any geometry. In general, the diabatic states can be described as linear combinations of adiabatic states (a subset or the complete set), though in practice they are frequently generated directly,4−7 rather than as a transformation of two or more adiabatic states. Diabatic states change their character slowly in the (nuclear) neighborhood of adiabatic state crossings and avoided crossings, and, particularly for weakly interacting © 2013 American Chemical Society

systems, this is an important and simplifying feature in treating the dynamics. The advantages of localized diabatic descriptions for electronic states are exploited in commonly used theories for electron transfer (ET)8 and excitation energy transfer (EET)9−11 between donor (D) and acceptor (A) sites, as well as in recent theories of proton-coupled electronic transfer (PCET) processes.12 Diabatic electronic states have been applied in many other studies of ground- and excited-state dynamics in the gas phase and in condensed phases.13−19 While the idealized formal definition is that they diagonalize the electronic matrix of the nuclear momentum operators, Mead and Truhlar20 showed that they cannot be defined globally except for a small subset of systems or nuclear coordinates. Hence, diabatization procedures generate at best quasi-diabatic states.7 As a result, one must choose between the results of a variety of methods, and it is of interest to compare the results obtained for different choices of diabatization schemes. A recent study21 of PCET based on a system with reduced nuclear dimensionality found excellent agreement between diabatic states obtained from minimization of derivative couplings and those given by the Special Issue: Kenneth D. Jordan Festschrift Received: September 5, 2013 Revised: November 20, 2013 Published: November 23, 2013 7221

dx.doi.org/10.1021/jp408913k | J. Phys. Chem. A 2014, 118, 7221−7234

The Journal of Physical Chemistry A

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Generalized Mulliken−Hush (GMH) method,22,23 a localization scheme that plays a central role in the present study. In formulating a localized diabatic model for electronic transport, one typically seeks states that are maximally localized on D and A sites, as well as intervening bridge (B) sites in, for example, DBA and DBBA assemblies. The precise degree of charge (or excitation) localization, corresponding, for example, to totally localized (“bare”) or partially delocalized (“dressed” with tails extending onto other sites), is controlled by the criterion defining the diabatic states (see section 2). In all cases, one arrives at a set of electronic states at each geometry that have diabatic state energies (diagonal elements of the Hamiltonian in the diabatic basis) and electronic coupling elements (off-diagonal elements of the Hamiltonian). To the extent that the diabatization scheme yields states for which the derivative coupling is actually small, the off-diagonal electronic coupling elements capture the dominant contributions to coupling on the diabatic surfaces.3,24 In the following discussion, the various diabatic state models under consideration are labeled for simplicity as bare or dressed depending on whether the delocalization in the state stems primarily from orthogonalization effects or also includes contributions from superexchange. Given the criterion of maximal localization, it must be emphasized that this is subject to the constraint of orthogonality, which unavoidably introduces some degree of delocalization. Like any observable quantity, the coupling elements Hij (e.g., HDA, whose square appears as a prefactor in the familiar two-state Golden Rule rate constant for ET) must be independent of an arbitrary zero of the operator (here, the Hamiltonian) and hence must be expressed in terms of orthogonal states.25 Thus, for example, if one employs initially nonorthogonal (NO) states, they must subsequently be orthogonalized either explicitly or implicitly, as discussed below in some detail. In section 4, the concept of bare states is examined in the context of the required orthogonality. In recent years, a number of diabatization schemes have arisen for the description of chemical processes. Among many options are (1) the method of Atchity and Ruedenberg,26−28 (2) the four-fold way of Nakamura and Truhlar,29,30 (3) the block-diagonalization (BD) method of Cederbaum, Domcke, and co-workers,3,13−15,31 (4) the GMH method,22,23,32 (5) the NO symmetry-broken Hartree−Fock method of Newton,4,33 (6) the fragment charge difference (FCD) method of Voityuk and Rösch,34 (7) the constrained density functional theory approach of Wu and Van Voorhis,6 (8) the approach of Warshel and co-workers based on valence-bond wave functions,17,18,35,36 (9) the recent methods developed by Subotnik9,37−39 and co-workers based on state-level extensions of the orbital localization approaches of Boys40 and Edmiston and Ruedenberg,41 (10) the tunneling currents approach of Stuchebrukhov and co-workers,42−45 (11) the neighboring orbital model (NM) of Nelsen and Zink,46 and (12) the frozen density embedding method of Pavanello and Neugebauer.7 Each has advantages and has been applied to important chemical problems. The utility of any diabatic state description can be assessed either by comparison with experimental results or with an accurate converged adiabatic description of the dynamics. Absent such comparisons, it is hard to argue for the “correctness” of a particular reduced space diabatic representation (i.e., one obtained for a subspace of n adiabatic states, selected so as to encompass the relevant D, B, and A sites).

However, for any given method, one still can ask about the (potential) physical meaning of the various parameters generated by the approach (diabatic state energies and offdiagonal Hamiltonian coupling elements). Some of the above methods can generate several diabatic states at once when more than two adiabatic states are chosen as the input reaction subspace.22,23,47−57 The multistate (i.e., n > 2) results have helped elucidate contributions from multiple ET processes occurring in a given DBA system32,48,49 and have highlighted the importance of certain bridge (and ligand) states in mediating the coupling in a series of analogous compounds.56 Finally, one can use the multistate couplings in a perturbative approach to “recover” two-state D−A couplings in a manner quite suggestive of superexchange treatments in model systems.56 It is to be expected that there could be significant differences between alternative diabatic representations even though they may seem to be based on similar physical criteria. Unfortunately, it has often been difficult to know how best to interpret the coupling elements arising from the various multistate treatments33,50,58,59 and especially how to compare matrix elements for a given DBA system based on different values of n. It may be tempting to believe that the most appropriate localized diabatic states correspond to fully localized bare states, such as one would encounter, for example, on D, unaffected by the presence of B or A, subject of course to the orthogonality constraint noted above (coupling between such bare states would be a candidate for so-called throughspace (TS) coupling, a topic discussed in section 4). Furthermore, there is some disagreement over the meaning of the multistate coupling elements themselves. That is, as n increases, do multistate treatments tend to an estimate of the DA coupling in the f ull DBA system, do they yield a restricted estimate of the couplings between localized states, with the impact of the other diabatic states removed (approximately or exactly), or is there another interpretation that better fits the calculated couplings? Also, in addressing overall DA coupling as n increases, one must deal with the influence of individual coupling elements for an increasing range of site−site contacts, from nearest neighbor (NN) to variable degrees of non-nearest neighbor (NNN). As multistate methods and their applications proliferate, the time is ripe for an analysis of some of the fundamental characteristics of their output. In particular, with the Marcus60 formulation of charge-transfer kinetics in mind, neglecting more general nuclear dynamics effects (see below), we focus on the above questions in a static (BO) framework. In this framework, we wish to address the following questions: (1) In treatments beyond that of a two-state model (n > 2), how should one interpret, for example, the DA coupling elements as the number of states increases? (2) Is it possible to use the multistate diabatic couplings to construct dressed two-state couplings, and are the multistate couplings transferable between different molecular environments? (3) How strongly do the couplings depend on the diabatization technique for methods possessing similar physical motivations (e.g., charge localization)? In the present article, we propose to address these questions in terms of concrete calculated results for various linear systems (DBA, DBBA, and DBB) using an extremely simple oneelectron model, a single electron in a potential comprised of one-dimensional (1-D) delta function potentials at each site. Of course, the answers to these questions based on the 1-D delta 7222



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function model (df m) may not apply directly to many-electron systems possessing more complicated potentials. However, if this simple model approach suggests that the interpretation of the multistate treatments is complex, it is unlikely that the situation will simplify greatly in systems with greater detail. We will show that, indeed, the simple model suggests multistate treatments to be much more complex than might be expected, and as a result, we conclude that many-electron multistate coupling elements need to be interpreted with caution. The following static BO treatment involves relatively large gaps between D/A and B levels and thus does not address the time-dependent behavior for gaps of intermediate magnitude, where the standard dichotomy between hopping and superexchange (labeled below as se) mechanisms breaks down.61−64 Also, relatively short DA separations are dealt with here ( VB) type (denoted hereafter, respectively, as e and h). Symmetry-equivalent D and A values would correspond to the transition state for thermal ET, but as a test of non-Condon (or “off shell”) behavior, examples of VD ≠ VA were also examined. The values VD = VA = −0.89 au and VB = −0.647 au (e) and −1.08 au (h), together with 3a0 separations between successive sites, are found (sections 3 and 4) to give results roughly comparable in magnitude to those from previous studies of DBA coupling.56,58,69,70 These choices of Va correspond to zeroth-order vertical diabatic energy gaps between degenerate D and A (denoted D/A) and B states (εB − εD/A) of equal magnitude (but opposite sign), ±0.187 h 7223



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Table 1. D/A Coupling (103 cm−1)a dressed (effective two-state) method

VB /au

model

“bare” (three-state (3/3/3))

partitioned three-state (augb)

basis set basis set basis set basis set exact exact exact exact

−0.647d −0.647d −1.080d,e −1.080e −0.647d −0.647d −1.080e −1.080e

LO GMH LO GMH BD/LO GMH BD/LO GMH

+2.85 +1.67 +1.15 +1.58 +5.70 +1.79 +0.88 +1.15

−3.24 (−6.09) −2.90 (4.57) +5.04 (+3.89) +4.91(+3.33) −5.30 (−11.0) −3.28 (−5.08) +4.47(+3.58) +4.40(+3.25)

3/3/2

(3/3/2−3/3/3)c

−2.38

−4.05

+4.41

+2.83

−2.60

−4.39

+3.94

+2.79

Bare (three-state (3/3/3)) and dressed (effective two-state), VD = VA = −0.89 au; ED = EA = −0.396 h. bThe aug denotes the augmentation (superexchange) term in eq 3. cThis quantity is the difference between the 3/3/3 and 3/3/2 coupling elements. dEB = −0.209 h (electron (e) transfer); D/A−B gap = +0.187 h. The cited gaps are zeroth order, based on the E values. The gaps in the 3/3/3 and 4/4/4 models (including basis set and exact results) are essentially the same (within 3%) as the zeroth-order results for h transfer and within 20% of the latter for e transfer. eVB = −1.080 au, EB = −0.583 h (hole (h) transfer); D/A−B gap = −0.187 h. a

the state diabatic dipole moment difference, and ΔXij  Xj − Xi.22) Many of the diabatization schemes discussed above can be implemented using the dfm approach. In the case where exact solutions (denoted “exact” below) for the adiabatic states are employed (as opposed to approximate solutions in terms of localized basis states), one can construct diabatic states and energies using (1) the GMH approach22,23 (based on the adiabatic state energies and dipole moments and transition moments) (2) a BD treatment13,23,72,73 where the NO localized singlesite eigenstates are projected onto the space of exact adiabatic states and the resulting states (still NO) are then symmetrically orthogonalized using the Lowdin procedure (LO). We denote this method as BD/LO. Unlike GMH, which depends only on a given state space, BD/LO states depend on the initially selected NO basis within the space. When the basis set (approximate) solutions of the Schrödinger equation are used, the following diabatization schemes can be adopted: (1) the GMH approach (2) a procedure that Löwdin othogonalizes74 the NO singlesite states and transforms the H matrix to this new diabatic basis (denoted LO). While this procedure bypasses the need for any explicit reference to the adiabatic states (in contrast to the standard BD method), the spirit of the method is identical to that of the actual BD approach. For brevity, we refer to it simply as the LO method. (3) We have explored other methods including the FCD34 and NM46 approaches. While the results for individual coupling elements vary with method (as they do for the GMH and LO results, vida infra), these methods yield similar answers to the three questions that we seek to address here (see above). For brevity’s sake, we thus omit further reference to these other methods. The GMH and BD/LO methods apply to general coupling situations and are not limited to the weak-coupling peturbative limit. We note that the degree of localization in both cases is determined by a variational procedure, with GMH maximizing the diabatic state centroid separations by diagonalizing the dipole matrix (based on vector components in the direction of the ET process)22,23 and with LO implementing a “least motion” criterion by minimizing the variance of differences between the initial NO and the orthogonal set.74

Partitioning Method. When 3/3/3 calculations are performed, it is possible to use a partitioning treatment75,76 to estimate the results of the dressed 3/3/2 calculation in the same potential. One chooses two states as a zeroth-order space (e.g., the bare 3/3/3 D and A states) and treats the third state (B) as a perturbative dressing of the interaction between the two bare states. The effective two-state coupling between D and A corresponds to se and can be written as56 (HDA )partitioned = HDA +

HDBHBA = HDA + (HDA )aug ΔE D/A,B (3)

where we adopt the commonly used mean value expression for the energy gap74,77 ΔE D/A,B = 0.5(HDD + HAA ) − HBB

(4)

with all quantities taken from the 3 × 3 H matrix for the method of interest (GMH or BD/LO) and with the total partitioned result displayed as the augmentation (aug) of the bare coupling.56 Extension of the partitioning model to the four-state case (DBBA) is considered in conjunction with the calculated results (section 3). Orthogonality. Some perspective on the role of orthogonalization is provided by considering the familiar two-state expression78 ′ = HDA

NO NO (HDA ) − SDAHD/A 2 (1 − SDA )

(5)

where the quantities on the rhs refer to a NO pair of states (D,A) and NO HD/A =

NO NO (HDD ) + HAA 2

(6)

(the NO superscript is suppressed in SDA because only the NO basis has a finite overlap element). While the overlap-adapted expression for HDA ′ has in the past been loosely referred to as the coupling of two NO states, ψD and ψA, in fact it is more properly recognized as the coupling of two orthogonal states, ψ′D and ψ′A, which, for the symmetric situation, can be expressed as the normalized ± combinations of the two eigenstates, ψ± formed from the NO states 7224



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Table 2. Bare Coupling (103 cm−1)ab VB/au

site

pair

model

−0.647 −0.647 −0.647 −0.647 −0.647 −0.647 −1.080 −1.080 −1.080 −1.080 −1.080 −1.080

NNN NNN NNN NNN NN NN NNN NNN NNN NNN NN NN

DA DA DB DB DB/BA DB/BA DA DA DB DB DB/BA DB/BA

LO GMH LO GMH LO GMH LO GMH LO GMH LO GMH

four-state

three-state

4/4/4

3/3/3

3/2/2

2/2/2

+2.85 +1.67 +2.39 +2.14 −16.7 −15.0 +1.15 +1.58 +0.92 +1.20 −12.4 −11.7

−1.42 −1.42 −2.79c −2.37 −15.2 −13.5 −1.82 −1.82 −1.33d −1.22 −12.2 −11.4

−0.83 −0.83 −1.79 −1.36 −15.6 −13.8 −0.83 −0.83 −0.70 −0.60 −12.2 −11.5

+2.63 +2.25 −16.7 −15.1

+0.95 +1.21 −12.3 −11.6

two-state

a DA and DB (next-nearest neighbor (NNN)) and DB,BA (nearest neighbor (NN)). bNNN sites separated by 6a0, with DA coupling based on a DBA triad and DB coupling based on a DBBA tetrad (4-state) and DBB triad (3-state and 2-state); NN sites (DB, BA) separated by 3a0, with coupling based on DBBA tetrad (4-state) and DBA triad (3-state and 2-state); due to symmetry, the DB and BA coupling elements are equal. The exact GMH and LO NN results are within 1% of the basis set results, with the exception of the LO e case, where exact NN coupling is 15% larger. c− 2.74 (4/2/2 1° LO estimate (see text (section 4)). d−1.34 (4/2/2 1° LO estimate (see text (section 4)).

Table 3. D/A Coupling in DBBA Assemblies (103 cm−1)a dressed (effective two-state) partitioned four-state (aug)b VB/au

model

“bare” (four-state (4/4/4))

−0.647 −0.647 −1.080 −1.080

LO GMH LO GMH

−0.29 −0.33 −0.10 −0.10

multiple pathwaysd −0.83 −0.68 −1.83 −1.77

single NN McConnell pathwaye −2.41 −1.99 −1.11 −0.95

(−0.54) (−0.35) (−1.73) (−1.67)

(−2.20) (−1.66) (−1.02) (−0.87)

4/4/2

4/4/2−4/4/4c

−0.46; β = 2.06/Åf

−0.13

−1.43; β = 1.42/Åf

−1.33

Bare (four-state (4/4/4)) and dressed (effective two-state); VD = VA = −0.89 au; ED = EA = −0.396 h (basis set results). bSee footnote a in Table 1. c The four-state analouge of (3/3/3−3/3/2); see footnote b in Table 1. dOn the basis of the full bridge Green’s function and including all NN and NNN tunneling pathways starting at site D (with no returns) and terminating with the first arrival at site A. eOn the basis of the simple forward NN McConnell tunneling pathway, D → B → B → A. fDefined as −(2 ln R)/δr, where R is the ratio of 4/4/2 and 3/3/2 DA coupling magnitudes and δr is the intersite separation (3 × 0.5292 Å). a

3. RESULTS Matrix elements (HDA) for electron (e) and hole (h) charge transfer in the DBA triad using both the exact and basis set treatments are presented in Table 1. There, we compare bare (3/3/3) and dressed (either partitioned 3/3/3 or, for the GMH case, 3/3/2 results). Table 2 displays results for bare (undressed) NNN and NN coupling (3/3/3, 3/2/2, and 2/ 2/2). The exact GMH and LO NN results are within 1% of the basis set results, with the exception of the LO e case, where exact NN coupling is 15% larger. The NNN DA cases correspond to DBA (3/3/3), D(B)A (3/2/2), and D(-)A (2/2/2), and similarly for NNN DB coupling, based on DBB, D(B)B, and D(-)B. Here, the notations (B) and (-) denote, respectively, the presence (but without an associated B basis state) and absence of the central B-site potential (VB). The NN DB couplings are based on DBA or DBB, DB(A) or DB(B), and DB(-), where now (B) and (-) refer to a terminal site (for DBA, HDB = HBA by symmetry). Comparing the 3/2/2 e and h results in Table 2, we see that for fixed site separations and D,A site potentials, as the central potential (VB) increases in magnitude, the magnitude of the DA coupling also increases. Table 3 offers the basis set four-site counterpart of Table 1 (for linear DBBA, with 3a0 NN site separations) and also includes calculated decay coefficients, β (Å), based on the relation β = −2 ln|(HDA(4/4/2)/HDA(3/3/2)|/δr, where δr = 3

′ = (ψ+ + ψ−)/21/2 ψD,A = ((ψD + ψA )/(2(1 + SDA ))1/2 ± (ψD − ψA ) /(2(1 − SDA ))1/2 )/21/2

(7)

We thus obtain a result equivalent to eq 5 ′ = HDA

(E+ − E−) 2

(8)

where E+ and E− are the two-state eigenvalues E± =

NO NO (HD/A ± HDA )

(1 ± SDA )

(9)

The analysis leading to eqs 7−9 applies to cases of NO degenerate HNO DD and HAA , either by virtue of symmetryequivalent D and A or otherwise accidental degeneracy or by using an arithmetic mean (eq 6). Through first order in overlap (HNO AA or SDA), when the denominator becomes unity, eq 5 coincides with the LO result, as discussed in section 4. The consequences of nonorthogonality for two-state ET dynamics have been discussed in detail by Efrima and Bixon79 and Migliore.80 7225



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× 0.5292 = 1.59 Å. The NN 4/4/4 and 4/2/2 coupling values are given in Table 2. Table 3 also displays the aug and partitioned results for the DBBA tetrad at two levels, based on the following: (1) inclusion of the full bridge Green function81 and all NN and NNN e or h tunneling pathways58,82 starting at D (with no returns) and ending with first arrival at A. (ET superexchange DA coupling, when based on perturbation theory (PT), is often described (refs 58, 59, 69, 70, 78, 81, and 82) as a superposition (with interference) of contributions from multiple “pathways”, where each pathway corresponds to a sequence of zeroth-order diabatic states, starting with the donor state (D), followed by a set of intermediate (virtual) bridge states, [Bl], and terminating with the acceptor state (A). For such a pathway, the PT yields a (p + 1)-order product of the form HDBiHBiBjHBjBk...HBlA/ΔEiΔEjΔEk...ΔEl, where ΔEi  EBi − ED/A (with D and A taken as degenerate) and i, j, k ... l as some sequence of bridge states involved in the pathway, p in number. The forward NN sequence defines the lowest-order (nbridge units + 1) McConnell pathway, where n is the number of units in a linear bridge. In general, p can range from 1 to infinity and may include backward “virtual hops” to bridge states already “visited” in the pathway.) (2) The simple forward NN McConnell tunneling pathway (D → B → B → A).83 The following discussion refers to the full bridge Green’s function treatment, except as noted otherwise. The magnitude of the dressed two-state DA coupling elements in Tables 1 and 3 indicate that for the DBA and DBBA examples considered in the present work, the ET rate constants will be in the adiabatic limit.8,24 Because the bare 3/3/3 and 4/4/4 HDA values are appreciable relative to the aug values (ratios range from 0.25 to 0.52 (three-state) and 0.06 to 0.94 (four-state), an alternative measure of se was obtained by subtracting the 3/3/3 (4/4/4) from the 3/3/2 (4/4/2) result and is included in Tables 1 and 3. Sizable ratios of the type noted here have also been found in related data in the literature.56,58,69,70,84 Further discussion of se and the interpretation of the bare coupling (3/3/3 and 4/4/4) are deferred to section 4. Additional results (not shown) were obtained for a larger intersite separation (5a0), and with somewhat different Va values (VD,A = −0.89 au and VB ranging from −0.5 to −1.5 au), yielding trends similar to those in Tables 1−3 (but with smaller coupling magnitudes due to weaker ovelap), including appreciable interference between bare (3/3/3) and superexchange (aug) terms. We also examined non-Condon effects by using different potentials for D and A, as discussed above (VD = −0.88 au, VA = −0.90 au). The results for the individual matrix elements in, for example, a 3/3/3 calculation were quite similar (within 0.5%) to those from the degenerate case, as were the dressed partitioned 3/3/3 and 3/3/2 results. This suggests that energetic non-Condon effects are not significant for this range of potential values. Trends in Calculated Coupling. Several consistent patterns can be seen in the calculated coupling magnitudes. For the dressed DA coupling, Tables 1 and 3 reveal the following trends, where the notation h > e and so forth denotes the relative magnitude of the h and e coupling (the bare NN and NNN couplings are taken up in section 4): (1) For aug and total partitioned terms, LO > GMH but are quite similar except for some of the three-state e transfer results, where GMH is less than LO by 40 (exact partitioned), 25 (basis set aug), and 54% (exact aug).

(2) For GMH, 3/3/2 < 3/3/3-partitioned results, and (3/3/ 2−3/3/3) < aug (smaller by 10−20%); for the four-site counterparts, the corresponding quantities are less by ∼20− 60%; exact and basis set three-state results are comparable; h > e for partitioned results; e > h for three-state aug and (3/3/2− 3/3/3); and h > e for four-state aug and (4/4/2−4/4/4). Comparison of Single and Multiple Tunneling Pathways for Four-Site DA Coupling. Table 3 reveals the strong interference among multiple tunneling pathways in comparison with the single McConnell pathway (especially for basis set results). The full (multiple pathway) partitioning method yields results in generally closer agreement with those from the 4/4/2 model than is the case when the McConnell approach is used.

4. DISCUSSION Comparison of NN and NNN Coupling and Assessment of Transferability. As part of the focus on two-state versus multistate coupling models, we consider the extent to which bare two-state (3/2/2 or 2/2/2) coupling elements resemble the bare three-state elements (3/3/3). If this were the case, it would allow the dressed three-state DA coupling (partitioned-3/3/3 or 3/3/2) to be evaluated solely in terms of bare two-state results (we continue with the shorthand l/m/n for sites/basis/states). A glance at Table 2 makes clear that the NN DB and BA elements (at 3a0 site separation) are quite transferable; that is, the bare 3/3/3, 3/2/2, and 2/2/2 results for a given method (GMH or LO) agree to within ±10%, whereas the NNN DA coupling (6a0 separation) for 3/3/3 and 3/2/2 differ appreciably in sign as well as magnitude (very similar behavior is observed for DB NNN coupling in the DBB triad). Essentially the same NN DB (and BA) coupling is found for DBA, DBB, and DBBA, irrespective of model (i.e., 3/3/3, 3/2/2, 4/4/4, and 4/2/2). While the DA two-atate coupling is sensitive to the presence (3/2/2) or absence (2/2/2) of the central B potential, the NN coupling is unaffected ( 2) related to two-state (n = 2) results (maintaining the same DA separation)? This question applies to the dressed (3/3/2 and 4/4/2) states. Related questions associated with these matrix elements include, (i) are the multistate DA coupling elements obtained here at the three-state (3/3/3) or four-state (4/4/4) levels “better” in some sense than the twostate results, (ii) which type of coupling is most physically appropriate for modeling ET kinetics in DBA systems and comparing with experimental data, and (iii) do the multistate HDA values (n > 2) capture only a portion of the overall DA coupling? A common assumption in the literature is that most coupling elements inferred from experimental kinetics correspond to dressed two-state coupling (i.e., dressed by partial delocalization of D and A onto B, as expressed by the variational GMH model or the perturbative partitioning treatment), and there is no indication from the present GMH calculations that an increase in n for the series l/l/n (i.e., where l = m ≥ n) leads to estimates that could be considered as appropriate superexchange-based multisite coupling elements for use in a twostate rate expression for ET. It is the dressed two-state coupling element (i.e., with n = 2) that allows one to place the ET process within the nonadiabatic and adiabatic limits. In fact, only when the partitioning approach is employed for the threeor four-state treatments does one recover a coupling element that adequately approximates the dressed two-state result. Indeed, the partitioned result is generally in strong contrast to the coupling of nominally bare D and A states (i.e., as given by 2/2/2, 3/2/2, 3/3/3, 4/2/2, and 4/4/4 levels). On the other hand, the nondressed multistate diabatic levels may be said to provide an increasingly fine-grained representation of the DBA electronic Hamiltonian, and this increased physical detail may be useful in modeling dynamics beyond the simple two-state kinetic framework. However, we conclude that if one’s aim is to calculate an estimate of the l/m/ 2 matrix element, the l/m/n matrix elements with n > 2 do not provide relevant estimates. Moreover, they are only indirectly connected to the bare 2/2/2 or l/2/2 matrix elements that one might consider at a zeroth-order level. (2). Construction of Effective Two-State Results from Multistate Coupling Elements and Transferability of Coupling Elements. The results of the partitioning treatment demonstrate that one can construct dressed two-state DA coupling in generally good agreement with the directly obtained GMH dressed two-state results (3/3/2 or 4/4/2). Note that this is dependent on the dressing state (B) being well-separated in energy and relatively weakly coupled to the D,A pair of states (as well as working in a regime of moderate charge-transfer distances and ignoring dynamical effects61,63,68,89). As to transferability, we have seen (Table 2) that while calculated NN coupling elements are quite similar at nondressed two-state and multistate (3/3/3) levels (i.e., they are “transferable”), the orthogonality effects arising from the intervening states render the NNN coupling elements non7230



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contributions from the bridge in a DBA system. This is clear, for example, from the near-equality (within 10−25%) of the 3/ 3/2 and partitioned 3/3/3 results. Exponential decay of twostate GMH superexchange coupling has been previously reported.91 However, these results suggest that the apportioning of various pieces of the two-state coupling between local sites obtained from multistate models (n > 2) does not necessarily correspond to physical expectations as a result of orthogonality, despite the fact that these diabatization procedures rest on clear physical criteria.

GMH results differ from those requiring the specification of Etunnel using a perturbative analysis. Applying partitioning theory for a symmetric DBA system and assuming localized (orthogonal) D, B, and A diabatic states, it is found (Appendix A in the Supporting Information) that the differences are quite small, being second-order in the quantity (HDB/(HBB − HDD)), assumed small relative to unity, with the magnitude of the perturbative partitioned result constituting an upper limit relative to the (exact) GMH result,77 as borne out by the dressed values reported in Tables 1 and 3. One can also compare on-shell and off-shell GMH DA coupling elements, in effect assessing the GMH behavior in a non-Condon situation. It turns out that at the two-state level treated in Appendix B (in the Supporting Information), the GMH coupling is invariant to changes in the splitting of HDD and HAA with all other Hamiltonian and dipole matrix held fixed, a result supported by calculations using the df m presented above (section 2). If HDA and the splitting of diabatic diagonal H elements (denoted δ) are defined to be independent, then obviously by construction, varying δ can have no effect on HDA. However, for the GMH model, this invariance is nontrivial because each of the three factors in HGMH (see refs 22 and 23) varies with δ, DA and yet, the overall coupling is found (by cancellation) to be independent of δ. This point is demonstrated in Appendix B (in the Supporting Information). More than One State Per Site. The present analysis does not directly address the use of multistate treatments in, for example, two-site, three-state models, which frequently occur in cases of photoexcited ET involving a ground state (GS), a locally excited state (LE), and a charge-transfer (CT) state.32,49 Here, under the assumption of local adiabaticity of the GS and LE states,22,56 the interactions of the GS and LE states with the CT state are not explicitly mediated by each other. To that extent, treatment of the GS−CT and LE−CT couplings is rather analogous to the 3/3/2 method above and thus yields couplings appropriate to the pairs of localized states of interest but influenced by the third state. Prospects and Caveats. Despite the seemingly complex results obtained here, we expect that multistate treatments, of necessity expressed in terms of orthogonal states, can play an important role in elucidating the coupling in DBA-type systems, with some caveats. Orthogonalization effects appear to have the biggest impact on NNN couplings, but for bridges with several states/pathways, it should still be feasible to compare couplings for different pairs of states localized on NN atoms. Multistate treatments can also be useful in comparing couplings mediated by different ligands. Finally, because the multistate treatment (in the perturbative limit and even for cases that do not appear to be candidates for PT) can still reproduce the effective twostate DA coupling elements via partitioning theory at a useful semiquantitative level (e.g., ∼10−25% for DBA in Table1 and less well (∼25−50%) for DBBA (Table 3)), such multistate approaches can provide an approximate means to correct for improper gaps by adjusting the energy denominator in the augmentation portion of the partitioned matrix element (eq 3). This would be particularly useful in one-electron treatments, where one might expect that the off-diagonal H matrix elements are reasonably accurate but where DB and AB gaps are inaccurate. We emphasize that the present results in no way imply that the two-state couplings obtained from, for example, the GMH method are not capable of accounting for superexchange

5. CONCLUSIONS Using a simple one-electron model, we investigated the performance of two physically motivated multistate treatments (GMH and BD/LO) for the ET matrix element. We found that the methods yield similar results for the partitioned effective two-state coupling elements between the D and A sites. However, the non-nearest-neighbor (NNN) multistate couplings do not correspond to “bare” coupling elements linking isolated localized diabatic states. Furthermore, the results also suggest that increasing the number of adiabatic states used in the diabatization scheme should not be considered to lead to improved estimates of coupling relative to the two-state matrix element but rather simply leads to a larger set of more localized (though still orthogonal) diabatic states.9 The overall two-state coupling can still be recovered through a partitioning treatment, but the physical interpretation of the orthogonal diabatic states from multistate treatments is neither trivial nor likely to become simple in more complex systems. In the future, the df m can be used to compare the results of the sorts of diabatization approaches used here with diabatic states obtained from rotations of the adiabatic states based on matrix elements of nuclear derivative coupling terms. This would allow a direct assessment of the strengths and weaknesses of various diabatization schemes.

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ASSOCIATED CONTENT

S Supporting Information *

Two appendices are included that deal with (i) the difference between GMH electronic couplings and perturbative treatments in DBA systems (Appendix A) and (ii) the invariance of the GMH coupling element to changes with respect to diabatic energy differences (Appendix B). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (R.J.C.). *E-mail: [email protected] (M.D.N.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS R.J.C. gratefully acknowledges financial support from the National Science Foundation (CHE-0353199) and from Harvey Mudd College. The Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences of the U.S. Department of Energy is gratefully acknowledged for funding the research carried out by M.D.N. through Grant DE-AC02-98CH10886. We are also grateful to the reviewers 7231



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(22) Cave, R. J.; Newton, M. D. Generalization of the Mulliken− Hush Treatment for the Calculation of Electron Transfer Matrix Elements. Chem. Phys. Lett. 1996, 249, 15. (23) Cave, R. J.; Newton, M. D. Calculation of Electronic Coupling Matrix Elements for Ground and Excited State Electron Transfer Reactions: Comparison of the Generalized Mulliken−Hush and Block Diagonalization Methods. J. Chem. Phys. 1997, 106, 9213. (24) Newton, M. D.; Sutin, N. Electron Transfer Reaction in Condensed Phases. Annu. Rev. Phys. Chem. 1985, 35, 437. (25) Newton, M. D. Formalisms for Electron-Exchange Kinetics in Aqueous-Solution and the Role of Ab initio Techniques in Their Implementation. Int. J. Quantum Chem. 1980, 17, 363. (26) Atchity, G. J.; Ruedenberg, K. Orbital Transformations and Configurational Transformations of Electronic Wavefunctions. J. Chem. Phys. 1999, 111, 2910. (27) Atchity, G. J.; Ruedenberg, K. A Local Understanding of the Quantum Chemical Geometric Phase Theorem in Terms of Diabatic States. J. Chem. Phys. 1999, 110, 4208. (28) Atchity, G. J.; Ruedenberg, K. Determination of Diabatic States through Enforcement of Configurational Uniformity. Theor. Chem. Acc. 1997, 97, 47. (29) Nakamura, H.; Truhlar, D. G. The Direct Calculation of Diabatic States Based on Configurational Uniformity. J. Chem. Phys. 2001, 115, 10353. (30) Nakamura, H.; Truhlar, D. G. Direct Diabatization of Electronic States by the Fourfold Way. Ii. Dynamical Correlation and Rearrangement Processes. J. Chem. Phys. 2002, 117, 5576. (31) Domcke, W.; Woywod, C.; Stengle, M. Diabatic CASSCF Orbitals and Wave-Functions. Chem. Phys. Lett. 1994, 226, 257. (32) Rust, M.; Lappe, J.; Cave, R. J. Multistate Effects in Calculations of the Electronic Coupling Element for Electron Transfer Using the Generalized Mulliken−Hush Method. J. Phys. Chem. A 2002, 106, 3930. (33) Newton, M. D. Quantum Chemical Probes of Electron Transfer Kinetics. Chem. Rev. 1991, 91, 767. (34) Voityuk, A. A.; Rosch, N. Fragment Charge Difference Method for Estimating Donor−Acceptor Electronic Coupling: Application to DNA π-Stacks. J. Chem. Phys. 2002, 117, 5607. (35) Wesolowski, T.; Muller, R. P.; Warshel, A. Ab Initio Frozen Density Functional Calculations of Proton Transfer Reactions in Solution. J. Phys. Chem. 1996, 100, 15444. (36) Hong, G. Y.; Rosta, E.; Warshel, A. Using the Constrained DFT Approach in Generating Diabatic Surfaces and Off Diagonal Empirical Valence Bond Terms for Modeling Reactions in Condensed Phases. J. Phys. Chem. B 2006, 110, 19570. (37) Subotnik, J. E.; Cave, R. J.; Steele, R. P.; Shenvi, N. The Initial and Final States of Electron and Energy Transfer Processes: Diabatization as Motivated by System−Solvent Interactions. J. Chem. Phys. 2009, 130, 234102. (38) Subotnik, J. E.; Vura-Weis, J.; Sodt, A. J.; Ratner, M. A. Predicting Accurate Electronic Excitation Transfer Rates via Marcus Theory with Boys or Edmiston−Ruedenberg Localized Diabatization. J. Phys. Chem. A 2010, 114, 8665. (39) Subotnik, J. E.; Yeganeh, S.; Cave, R. J.; Ratner, M. A. Constructing Diabatic States from Adiabatic States: Extending Generalized Mulliken−Hush to Multiple Charge Centers with Boys Localization. J. Chem. Phys. 2008, 129, 244101. (40) Boys, S. F. In Quantum Theory of Atoms, Molecules, and the Solid State; Lowdin, P., Ed.; Academic Press: New York, 1966; p 253. (41) Edmiston, C.; Ruedenberg, K. Localized Atomic and Molecular Orbitals. Rev. Mod. Phys. 1963, 35, 457. (42) Stuchebrukhov, A. Ab Initio Calculations of Long-Distance Electron Tunneling in Proteins: Tunneling Currents Approach. J. Inorg. Biochem. 2001, 86, 101. (43) Stuchebrukhov, A. A. Tunneling Currents in Long-Distance Electron Transfer Reactions. III. Many-Electron Formulation. J. Chem. Phys. 1998, 108, 8499. (44) Stuchebrukhov, A. A. Tunneling Currents in Long-Distance Electron Transfer Reactions. IV. Many-Electron Formulation. Non-

for careful reading and helpful suggestions that both clarified the manuscript and suggested new avenues for investigation.

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REFERENCES

(1) Herzberg, G. Molecular Spectra and Molecular Structure; D. Van Nostrand: Princeton, NJ, 1967. (2) Longuet-Higgins, H. Some Recent Developments in the Theory of Molecular Energy Levels. Adv. Spectrosc. 1961, 2, 429. (3) Koppel, H.; Domcke, W.; Cederbaum, L. S. Multimode Molecular-Dynamics Beyond the Born−Oppenheimer Approximation. Adv. Chem. Phys. 1984, 57, 59. (4) Logan, J.; Newton, M. D. Ab Initio Study of Electronic Coupling in the Aqueous Fe+2−Fe+3 Electron Exchange Process. J. Chem. Phys. 1983, 78, 4086. (5) Cave, R. J.; Baxter, D. V.; Goddard, W. A. I.; Baldeschwieler, J. D. Theoretical Studies of Electron Transfer in Metal Dimers: Xy+ → X +Y, X,Y=Be, Mg, Ca, Zn, Cd. J. Chem. Phys. 1987, 87, 926. (6) Wu, Q.; Van Voorhis, T. Extracting Electron Transfer Coupling Elements from Constrained Density Functional Theory. J. Chem. Phys. 2006, 125, 164105. (7) Pavanello, M.; Neugebauer, J. Modelling Charge Transfer Reactions with the Frozen Density Embedding Formalism. J. Chem. Phys. 2011, 135, 234103. (8) Marcus, R. A.; Sutin, N. Electron Transfers in Chemistry and Biology. Biochim. Biophys. Acta 1985, 811, 265. (9) Vura-Weis, J.; Newton, M. D.; Wasielewski, M. R.; Subotnik, J. E. Characterizing the Locality of Diabatic States for Electronic Excitation Transfer by Decomposing the Diabatic Coupling. J. Phys. Chem. C 2010, 114, 20449. (10) Dexter, D. L. A Theory of Sensitized Luminescence in Solids. J. Chem. Phys. 1953, 21, 836. (11) Foerster, T. In Modern Quantum Chemistry, Istanbul Lectures III: Action of Light and Organic Crystals; OSinanoglu, O., Ed.; Academic Press: New York, 1965; p 93. (12) Hammes-Schiffer, S.; Stuchebrukhov, A. A. Theory of Coupled Electron and Proton Transfer Reactions. Chem. Rev. 2010, 110, 6939. (13) Domcke, W.; Woywod, C. Direct Construction of Diabatic States in the Casscf Approach  Application to the Conical Intersection of the 1A2 and 1B1 Excited-States of Ozone. Chem. Phys. Lett. 1993, 216, 362. (14) Krawczyk, R. P.; Malsch, K.; Hohlneicher, G.; Gillen, R. C.; Domcke, W. 1 1Bu1−2 1Ag Conical Intersection in trans-Butadiene: Ultrafast Dynamics and Optical Spectra. Chem. Phys. Lett. 2000, 320, 535. (15) Ostojic, B.; Domcke, W. Ab Initio Investigation of the Potential Energy Surfaces Involved in the Photophysics of S-trans-1,3-Butadiene. Chem. Phys. 2001, 269, 1. (16) Sudholt, W.; Sobolewski, A. L.; Domcke, W. Ab Initio Study of the Amino Group Twisting and Wagging Reaction Paths in the Intramolecular Charge Transfer of 4-(N,N-Dimethylamino)Benzonitrile. Chem. Phys. 1999, 240, 9. (17) Kamerlin, S. C. L.; Cao, J.; Rosta, E.; Warshel, A. On Unjustifiably Misrepresenting the Evb Approach while Simultaneously Adopting It. J. Phys. Chem. B 2009, 113, 10905. (18) Warshel, A.; Parson, W. W. Dynamics of Biochemical and Biophysical Reactions: Insight from Computer Simulations. Q. Rev. Biophys. 2001, 34, 563. (19) Werner, H. J.; Meyer, W. MCSCF Study of the Avoided Curve Crossing of the Two Lowest 1Σ+ States of LiF. J. Chem. Phys. 1981, 74, 5802. (20) Mead, C. A.; Truhlar, D. G. Conditions for the Definition of a Strictly Diabatic Electronic Basis for Molecular-Systems. J. Chem. Phys. 1982, 77, 6090. (21) Sirjoosingh, A.; Hammes-Schiffer, S. Diabatization Schemes for Generating Charge-Localized Electron−Proton Vibronic States in Proton-Coupled Electron Transfer Systems. J. Chem. Theory Comput. 2011, 7, 2831. 7232



Article

orthogonal Atomic Basis Sets and Mulliken Population Analysis. J. Chem. Phys. 1998, 108, 8510. (45) Stuchebrukhov, A. A. Dispersion-Relations for Electron and Hole Transfer in Donor Bridge Acceptor Systems. Chem. Phys. Lett. 1994, 225, 55. (46) Nelsen, S. F.; Weaver, M. N.; Luo, Y.; Lockard, J. V.; Zink, J. I. Use of the Neighboring Orbital Model for Analysis of Electronic Coupling in Class III Intervalence Compounds. Chem. Phys. 2006, 324, 195. (47) Zimmt, M. B.; Waldeck, A. H. Exposing Solvent’s Roles in Electron Transfer Reactions: Tunneling Pathway and Solvation. J. Phys. Chem. A 2003, 107, 3580. (48) Cave, R. J.; Newton, M. D.; Kumar, K.; Zimmt, M. B. Theoretical-Study of Solvent Effects on the Electronic Coupling Matrix Element in Rigidly Linked Donor−Acceptor Systems. J. Phys. Chem. 1995, 99, 17501. (49) Lappe, J.; Cave, R. J.; Newton, M. D.; Rostov, I. A Theoretical Investigation of Charge Transfer in Several Substituted Acridinium Ions. J. Phys. Chem. B 2005, 109, 6610. (50) Voityuk, A. A. Electronic Couplings in DNA π-Stacks: Multistate Effects. J. Phys. Chem. B 2005, 109, 17917. (51) Voityuk, A. A. Estimation of Electronic Coupling in π-Stacked Donor−Bridge−Acceptor Systems: Correction of the Two-State Model. J. Chem. Phys. 2006, 124, 064505. (52) Voityuk, A. A. Donor−Acceptor Electronic Couplings in πStacks: How Many States Must Be Accounted For? Chem. Phys. Lett. 2006, 422, 15. (53) Lambert, C.; Noll, G. The Class II/III Transition in Triarylamine Redox Systems. J. Am. Chem. Soc. 1999, 121, 8434. (54) Lambert, C.; Noll, G. Optically and Thermally Induced Electron Transfer Pathways in Hexakis[4-(N,N-diarylamino)phenyl]benzene Derivatives. Chem.Eur. J. 2002, 8, 3467. (55) Lambert, C.; Noll, G.; Schelter, J. Bridge-Mediated Hopping or Superexchange Electron-Transfer Processes in Bis(triarylamine) Systems. Nat. Mater. 2002, 1, 69. (56) Cave, R. J.; Edwards, S. T.; Kouzelos, J. A.; Newton, M. D. Reduced Electronic Spaces for Modeling Donor/Acceptor Interactions. J. Phys. Chem. B 2010, 114, 14631. (57) Pheasant, S.; Kouzelos, J. A.; Van Ryswyk, H.; Cave, R. J. Theoretical Investigation of the Electronic Coupling Element in BisRuthenium Porphyrin Dimers. Mol. Simul. 2006, 32, 677. (58) Liang, C.; Newton, M. D. Ab Initio Studies of Electron Transfer 2. Pathway Analysis for Homologous Organic Spacers. J. Phys. Chem. 1993, 97, 3199. (59) Liang, C.; Newton, M. D. Ab Initio Studies of Electron Transfer: Pathway Analysis of Effective Transfer Integrals. J. Phys. Chem. 1992, 96, 2855. (60) Marcus, R. A. On the Theory of Electron Transfer Reactions VI. Unified Treatment for Homgeneous and Electrode Reactions. J. Chem. Phys. 1965, 43, 679. (61) Sumi, H.; Kakitani, T. Electron Transfer via a Midway Molecule as Seen in Primary Processes in Photosynthesis; A New Process Describable as Superexchange or Sequential in Mutually Opposite Limits. Chem. Phys. Lett. 1996, 252, 85. (62) Sumi, H.; Kakitani, T. Unified Theory on Rates for Electron Transfer Mediated by a Midway Molecule, Bridging between Superexchange and Sequential Processes. J. Phys. Chem. B 2001, 105, 9603. (63) Saito, K.; Sumi, H. Unified Expression for the Rate Constant of the Bridged Electron Transfer Derived by Renormalization. J. Chem. Phys. 2009, 131, 134101. (64) Skourtis, S. S.; Archontis, G.; Xie, Q. Electron Transfer through Fluctuating Bridges: On the Validity of the Superexchange Mechanism and Time-Dependent Tunneling Matrix Elements. J. Chem. Phys. 2001, 115, 9444. (65) Beratan, D. N.; Hopfield, J. J. Failure of the Born−Oppenheimer and Franck−Condon Approximations for Long-Distance ElectronTransfer Rate Calculations. J. Chem. Phys. 1984, 81, 5753.

(66) Beratan, D. N.; Onuchic, J. N.; Hopfield, J. J. Some Aspects of Electron-Transfer Reaction Dynamics. J. Phys. Chem. 1986, 90, 3707. (67) Hwang, J. K.; Warshel, A. Microscopic Examination of FreeEnergy Relationships for Electron-Transfer in Polar-Solvents. J. Am. Chem. Soc. 1987, 109, 715. (68) Townsend, J. S. Quantum Physics, A Fundamental Approach to Modern Physics; University Science Books: Sausolito, CA, 2010; p 130. (69) Curtiss, L. A.; Naleway, C. A.; Miller, J. R. Chem. Phys. 1993, 176, 387. (70) Shephard, M. J.; Paddon-Row, M. N.; Jordan, K. D. Electronic Coupling through Saturated-Hydrocarbon Bridges. Chem. Phys. 1993, 176, 289. (71) Newton, M. D. Bridge-Mediated Electron Transfer and Multiple Reaction Coordinates. Isr. J. Chem. 2004, 44, 83. (72) Pacher, T.; Cederbaum, L. S.; Koppel, H. Approximately Diabatic States from Block Diagonalization of the Electronic Hamiltonian. J. Chem. Phys. 1988, 89, 7367. (73) Pacher, T.; Koppel, H.; Cederbaum, L. S. Quasidiabatic States from Abinitio Calculations by Block Diagonalization of the Electronic Hamiltonian  Use of Frozen Orbitals. J. Chem. Phys. 1991, 95, 6668. (74) Lowdin, P. O. On the Non-Orthogonality Problem Connected with the Use of Atomic Wave Functions in the Theory of Molecules and Crystals. J. Chem. Phys. 1950, 18, 365. (75) Larsson, S. Electron-Transfer in Chemical and BiologicalSystems  Orbital Rules for Non-Adiabatic Transfer. J. Am. Chem. Soc. 1981, 103, 4034. (76) Lowdin, P. O. Studies in Perturbation Theory. 1. An Elementary Iteration-Variation Procedure for Solving Schrodinger Equation by Partitioning Technique. J. Mol. Spectrosc. 1963, 10, 12. (77) Evenson, J. W.; Karplus, M. Effective Coupling in Bridged Electron-Transfer Molecules  Computational Formulation and Examples. J. Chem. Phys. 1992, 96, 5272. (78) Newton, M. D. Quantum Chemical Probes of Electron-Transfer Kinetics  The Nature of Donor−Acceptor Interactions. Chem. Rev. 1991, 91, 767. (79) Efrima, S.; Bixon, M. Outer-Sphere Electron-Transfer Reactions in Polar-Solvents. J. Chem. Phys. 1976, 64, 3639. (80) Migliore, A. Nonorthogonality Problem and Effective Electronic Coupling Calculation: Application to Charge Transfer in π-Stacks Relevant to Biochemistry and Molecular Electronics. J. Chem. Theory Comput. 2011, 7, 1712. (81) Ratner, M. A. Bridge-Assisted Electron-Transfer  Effective Electronic Coupling. J. Phys. Chem. 1990, 94, 4877. (82) Onuchic, J. N.; Beratan, D. N. A Predictive Theoretical Model for Electron-Tunneling Pathways in Proteins. J. Chem. Phys. 1990, 92, 722. (83) McConnell, H. M. Intramolecular Charge Transfer in Aromatic Free Radicals. J. Chem. Phys. 1961, 35, 508. (84) Paddon-Row, M. N.; Wong, S. S.; Jordan, K. D. A Cautionary Comment on the Use of Orthogonal Localized Molecular-Orbitals for the Quantitative-Analysis of Through-Space and Through-Bond Orbital Interactions. J. Chem. Soc., Perkin Trans. 2 1990, 425. (85) Paddon-Row, M. N. Some Aspects of Orbital Interactions through Bonds  Physical and Chemical Consequences. Acc. Chem. Res. 1982, 15, 245. (86) Verhoeven, J. W.; Pasman, P. The Relative Sign of ThroughBond and Through-Space Interactions  Sigma-Assistance of Cyclization and Intramolecular Hydrogen Transfer. Tetrahedron 1981, 37, 943. (87) Paddon-Row, M. N.; Wong, S. S.; Jordan, K. D. Ab Initio SCF MO and Natural Bond Orbital Studies of 7-Silanobornadiene and 7,7Dimethyl-7-silanobornadiene. Two Molecules Possessing an Inverted Sequence of π Orbitals. J. Am. Chem. Soc. 1990, 112, 1710. (88) King, G.; Warshel, A. Investigation of the Free-Energy Functions for Electron-Transfer Reactions. J. Chem. Phys. 1990, 93, 8682. (89) Sakata, Y.; Tsue, H.; Goto, Y.; Misumi, S.; Asahi, T.; Nishikawa, S.; Okada, T.; Mataga, N. Observation of Large Orientation Effect on 7233



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Photoinduced Electron-Transfer in Rigid Porphyrin−Quinone Compounds. Chem. Lett. 1991, 1307. (90) Onuchic, J. N.; Beratan, D. N.; Hopfield, J. J. Some Aspects of Electron-Transfer Reaction Dynamics. J. Phys. Chem. 1986, 90, 3707. (91) Henderson, T. M.; Cave, R. J. An Ab Initio Study of Specific Solvent Effects on the Electronic Coupling Element in Electron Transfer Reactions. J. Chem. Phys. 1998, 109, 7414.

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